Integration Hints

Date: 02/23/99 at 19:08:03
From: Scott
Subject: Question on Integration
Can you help me integrate:
(cos[x])^4
I have spent a full hour and I keep getting stuck, at a different point
each time, with a different answer. I think you have to split it up and
use identities instead, but I do not know which ones to use.
If you could help me out with this, by writing me either the work, or
a step-by-step proccess of how to solve this, I would GREATLY
appreciate it.

Date: 02/23/99 at 19:37:16
From: Doctor Wilkinson
Subject: Re: Question on Integration
I am not going to do all the work, but I can give you some useful
hints. There are a lot of ways to go about this, but one of the
simplest is to start by using the identity sin^2 x + cos^2 x = 1, and
rewriting the integral as the integral of
cos^2 x cos^2 x dx = cos^2 x (1 - sin^2 x) dx,
which is the integral of
cos^2 x dx - the integral of sin^2 x cos^2 x dx.
Next I would use the double-angle formulae
cos(2x ) = cos^2 x - sin^2 x = 2 cos^2 x - 1
to express the first integral in terms of the integral of cos (2x) dx
and
sin(2x) = 2 sinx cos x
to express the second integral in terms of the integral of sin^2(2x),
which again can be expressed in terms of the integral of cos(2x) dx by
using the same double-angle formula for the cosine in the form
cos(2x) = 1 - 2 sin^2 x.
I hope this helps. You have to proceed carefully and watch the signs
and so on.
- Doctor Wilkinson, The Math Forum
http://mathforum.org/dr.math/